How - if at all - does the integral $\int_{0}^{1} \Big{(} \frac{\ln(1-x)}{x} \Big{)}^{n} dx $ arise in Feynman diagrams?

Question

In the following table by A. Devoto and D. W. Duke, a number of logarithmic integrals are listed that - according to the authors - help with Feynman diagram calculations. In particular, integral (3.6.1) on p. 14 reads as follows: $$I := \int_{0}^{1} \frac{\ln(1-y)}{y} dy = - \zeta(2). $$ Here, $\zeta(\cdot)$ is the Riemann zeta function - and we have $\zeta(2) = \frac{\pi^{2}}{6}$.

Now, I am not a physicist, but a mathematician. I don't know how or when this integral exactly arises in Feynman diagrams and in quantum electrodynamics in general. My interest in this integral is mostly related to the following question I have on a particular quadratic rational zeta series. In that regard, I'm looking for integral expressions that amount to $$S_{k} := \sum_{i=1}^{k} \left[ {k \atop i} \right] \zeta(i+1). \tag{*} $$ In the above sum, the $\left[ {k \atop i} \right]$ are the unsigned Stirling numbers of the first kind.

Interestingly, we also have: \begin{align} I_{k} &:= \int_{0}^{1} \bigg{(} - \frac{\ln(1-y)}{y} \bigg{)}^{k} dy \\ &= k\sum_{n=0}^{k-1}\left[ k-1 \atop n\right]\zeta(k+1-n) . \tag{**}\end{align}

This expression comes close to $(*)$, though it is not quite the same. I'm currently looking for similar integrals that actually amount to $(*)$. I've already looked into numerous other integrals with powers of the integrand - often with logarithmic terms - but to no avail so far. In parallel to continuing on this path, I thought perhaps one could explore finding such an expression indirectly through Feynman diagrams.

My reasoning is as follows: if one could associate a Feynman diagram with expressions of the form $I_{k}$, then perhaps one could infer what kind of Feynman diagram should be associated with $S_{k}$, which in turn could yield information about the integral form that corresponds to $(*)$.

Questions:

1. How does the integral $I$ arise in Feynman diagram calculations? Is there a unique Feynman diagram that corresponds exactly to this integral?
2. Do integrals of the form $I_{k}$ also arise in Feynman diagram calculations? If so, how? Can one discern a pattern in the successive Feynman diagrams that are connected to $I_{1}, I_{2}, I_{3}, \dots$ ?
3. Is it also possible to associate a Feynman diagram to values - in particular of the form $S_{k}$ - or do they only correspond to the integrals themselves?

Answer 1

For what it's worth, multiple polylogarithms (MPLs), similar to OP's title example, appear in multiloop Feynman diagrams of already scalar theories, see e.g. Ref. 1 for details.

References:

1. C. Duhr, Mathematical aspects of scattering amplitudes, TASI lectures 2014, arXiv:1411.7538.